Second-order gradient expansion

With an empirical value of /3, neither B86 nor mB86 recovers the correct second-order gradient expansion of Eq. (81). DePristo and Kress [127] explicitly imposed this constraint by using the formula... 

The T/ recovers the first constraint by itself, but does not. In order to recover the correct second-order gradient expansion for the correlation energy in the s — 0 limit, another term is used ... 

For a density that is everywhere slowly varying, the exchange energy has a known second-order gradient expansion [27,28], which discards the 0(V ) terms in... 

Is There a Second-Order Gradient Expansion for the Slowly Varying Region ... 

If there was a second-order gradient expansion for the exchange energy density, it would have to take the form [20] of the zero-th and second-order terms in... 

As a proof-of-principle for our fitting, we show in Appendix 1 that we can fit the exact positive noninteracting kinetic energy density of the Airy gas to a known second-order gradient expansion. 

To see if there is a second-order gradient expansion for the conventional exchange-correlation energy density in RPA, we have fitted B in... 

For completeness, we have also looked for a second-order gradient expansion of the conventional correlation energy density, by fitting C in... 

Start with a density that is so slowly varying that the second-order gradient expansions are valid for Ex (Eq. (16)) and for Exc (Eq. (20)). This requires that the reduced density gradients on both length scales of Section 5 must be small. Thus p and q of Eqs. (10) and (11) have magnitudes much less than 1, and so do pc = k.F/ksf p and = kp/ksfq- Moreover, the second-order contribution to the correlation energy... 

We return to the task of setting the three parameters, p, k, and a, in Equation 14.13. We assume first that the correct value of the coefficient of the second-order gradient expansion in the limit 5 0 is the one that arises from the asymptotic expansion of the semiclassic neutral atom. Therefore, from Equation 14.14, one has V MGEA = P + 1) = 0.26. The second constraint we enforce is freedom from one-... 

There are two answers to the seeming paradox of the previous paragraph. The first is that realistic electron densities are not very close to the slowly-varying limit (s -C 1, p/s exchange-correlation hole is the hole of a possible physical system, the uniform electron gas, and so satisfies many exact constraints, as discussed in Sect. 1.6.1. The second-order gradient expansion or GEA approximation to the hole is not, and does not. 

In 1968, Ma and Brueckner [69] derived the second-order gradient expansion for the correlation energy in the high-density limit, (1.188) and (1.191). In numerical tests, they found that it led to improperly positive correlation energies for atoms, because of the large size of the positive gradient term. As a remedy, they proposed the first GGA,... 

In 1980, Langreth and Perdew [83] explained the failure of the second-order gradient expansion (GEA) for E. They made a complete wavevector analysis of be., they replaced the Coulomb interaction /u in (1.100) by its Fourier transform and found... 

To understand this pattern [21], we note that the second-order gradient expansion for the non-interacting kinetic energy Ts[n -, n, which is arguably its own GGA [81], can be written as... 

---